标题： 基于图的无网格形状优化使用神经网络和偏微分方程 显示英文标题

标题： Meshless Shape Optimization using Neural Networks and Partial Differential Equations on Graphs

摘要： 形状优化涉及在一组形状上定义的成本函数的最小化，通常由偏微分方程（PDE）控制。 在没有闭合解的情况下，人们依赖数值方法来近似求解。 水平集方法——当与有限元方法结合时——是最通用的数值形状优化方法之一，但仍受到大多数基于网格方法的限制。 在本工作中，我们提出了一种完全无网格的水平集框架，该框架利用神经网络对水平集函数进行参数化，并使用图拉普拉斯算子来近似底层的PDE。 我们的方法能够精确计算几何量，如表面法线和曲率，并允许在凸形状类中解决优化问题。 显示英文摘要

摘要： Shape optimization involves the minimization of a cost function defined over a set of shapes, often governed by a partial differential equation (PDE). In the absence of closed-form solutions, one relies on numerical methods to approximate the solution. The level set method -- when coupled with the finite element method -- is one of the most versatile numerical shape optimization approaches but still suffers from the limitations of most mesh-based methods. In this work, we present a fully meshless level set framework that leverages neural networks to parameterize the level set function and employs the graph Laplacian to approximate the underlying PDE. Our approach enables precise computations of geometric quantities such as surface normals and curvature, and allows tackling optimization problems within the class of convex shapes.